iation 7. Let f be continuous on [a, b] and differentiable on (a, b). If lim f'(x) xia exists in R, show that f is differentiable at a and f'(a) = lim f'(x). A similar result holds for b. x-a 8. In reference to Corollary 5.4, give an example of a uniformly continuous function on [0, 1] that is differentiable on (0, 1] but whose derivative is not bounded there. 9. Recall that a fixed point of a function f is a point c such that f(c) = c. (a) Show that if f is differentiable on R and f'(x)|< 1 for all x in R, then f has at most one fixed point. (b) Let f(x) = x + (1 + e)-1. Show that f satisfies the hypothesis of part (a) but that f has no fixed point. 10. Show that sin x > x if x < 0. 11. Show that (x-1)/x < Inx 1 and hence In(1+x) 0. 12. For 0 < x л/2. (Thus, as x л/2 from the left, cos x is never large enough for x+cosx to be greater than л/2 and cot x is never small enough for x + cot x to be less than x/2.)
iation 7. Let f be continuous on [a, b] and differentiable on (a, b). If lim f'(x) xia exists in R, show that f is differentiable at a and f'(a) = lim f'(x). A similar result holds for b. x-a 8. In reference to Corollary 5.4, give an example of a uniformly continuous function on [0, 1] that is differentiable on (0, 1] but whose derivative is not bounded there. 9. Recall that a fixed point of a function f is a point c such that f(c) = c. (a) Show that if f is differentiable on R and f'(x)|< 1 for all x in R, then f has at most one fixed point. (b) Let f(x) = x + (1 + e)-1. Show that f satisfies the hypothesis of part (a) but that f has no fixed point. 10. Show that sin x > x if x < 0. 11. Show that (x-1)/x < Inx 1 and hence In(1+x) 0. 12. For 0 < x л/2. (Thus, as x л/2 from the left, cos x is never large enough for x+cosx to be greater than л/2 and cot x is never small enough for x + cot x to be less than x/2.)
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Question
![iation
7. Let f be continuous on [a, b] and differentiable on (a, b). If lim f'(x)
xia
exists in R, show that f is differentiable at a and f'(a) = lim f'(x). A
similar result holds for b.
x-a
8. In reference to Corollary 5.4, give an example of a uniformly continuous
function on [0, 1] that is differentiable on (0, 1] but whose derivative is not
bounded there.
9. Recall that a fixed point of a function f is a point c such that f(c) = c.
(a) Show that if f is differentiable on R and f'(x)|< 1 for all x in R,
then f has at most one fixed point.
(b) Let f(x) = x + (1 + e)-1. Show that f satisfies the hypothesis of
part (a) but that f has no fixed point.
10. Show that sin x > x if x < 0.
11. Show that (x-1)/x < Inx <x-1 for x > 1 and hence In(1+x) <x
for x > 0.
12. For 0 < x <y, show that 1 - (x/y) <lny - Inx < (y/x) - 1.
13. For 0 < x <л/2, show that x + cos x <π/2 and x+cot x >л/2. (Thus,
as x л/2 from the left, cos x is never large enough for x+cosx to be
greater than л/2 and cot x is never small enough for x + cot x to be less
than x/2.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f89dc84-5360-495d-a77a-10dc4ac0ca49%2Ff7c88e1e-0afe-4847-92a9-2426e3b65239%2Fxynitfs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:iation
7. Let f be continuous on [a, b] and differentiable on (a, b). If lim f'(x)
xia
exists in R, show that f is differentiable at a and f'(a) = lim f'(x). A
similar result holds for b.
x-a
8. In reference to Corollary 5.4, give an example of a uniformly continuous
function on [0, 1] that is differentiable on (0, 1] but whose derivative is not
bounded there.
9. Recall that a fixed point of a function f is a point c such that f(c) = c.
(a) Show that if f is differentiable on R and f'(x)|< 1 for all x in R,
then f has at most one fixed point.
(b) Let f(x) = x + (1 + e)-1. Show that f satisfies the hypothesis of
part (a) but that f has no fixed point.
10. Show that sin x > x if x < 0.
11. Show that (x-1)/x < Inx <x-1 for x > 1 and hence In(1+x) <x
for x > 0.
12. For 0 < x <y, show that 1 - (x/y) <lny - Inx < (y/x) - 1.
13. For 0 < x <л/2, show that x + cos x <π/2 and x+cot x >л/2. (Thus,
as x л/2 from the left, cos x is never large enough for x+cosx to be
greater than л/2 and cot x is never small enough for x + cot x to be less
than x/2.)
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